What is (\partial)/(\partial x)(xsqrt(x^2+y^2+z^2)) ?

The answer to (\partial)/(\partial x)(xsqrt(x^2+y^2+z^2)) is (2x^2+y^2+z^2)/(sqrt(x^2+y^2+z^2))

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(\partial)/(\partial x)(xsqrt(x^2+y^2+z^2))

\frac{\partial}{\partial x} (x x^{2} + y^{2} + z^{2})

\frac{2 x ^{2} + y ^{2} + z ^{2}}{x ^{2} + y ^{2} + z ^{2}}

Solution steps

\frac{\partial}{\partial x} (x x^{2} + y^{2} + z^{2})

Treat

y, z

as constants

Apply the Product Rule:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = x, g = x^{2} + y^{2} + z^{2}

= \frac{\partial}{\partial x} (x) x^{2} + y^{2} + z^{2} + \frac{\partial}{\partial x} (x^{2} + y^{2} + z^{2}) x

\frac{\partial}{\partial x} (x) = 1

\frac{\partial}{\partial x} (x^{2} + y^{2} + z^{2}) = \frac{x}{x ^{2} + y ^{2} + z ^{2}}

= 1 \cdot x^{2} + y^{2} + z^{2} + \frac{x}{x ^{2} + y ^{2} + z ^{2}} x

Simplify

1 \cdot x^{2} + y^{2} + z^{2} + \frac{x}{x ^{2} + y ^{2} + z ^{2}} x : \frac{2 x ^{2} + y ^{2} + z ^{2}}{x ^{2} + y ^{2} + z ^{2}}

= \frac{2 x ^{2} + y ^{2} + z ^{2}}{x ^{2} + y ^{2} + z ^{2}}

\frac{d}{d x} (\frac{2}{4 - cos ( 2 x )})

\int \frac{1}{t ln ( t ^{2} )} d t

\int \frac{x ^{3} + 5}{5 x + 3} d x

\frac{d}{d x} (\frac{100 x}{x + 1})

\int (1 + e^{2 x}) sin (3 x) d x

What is (\partial)/(\partial x)(xsqrt(x^2+y^2+z^2)) ?
The answer to (\partial)/(\partial x)(xsqrt(x^2+y^2+z^2)) is (2x^2+y^2+z^2)/(sqrt(x^2+y^2+z^2))